Laminar drag reduction in surfactant-contaminated superhydrophobic channels*

Although superhydrophobic surfaces (SHSs) show promise for drag reduction (DR) applications, their performance can be compromised by traces of surfactant that accumulate at liquid-gas interfaces, generating Marangoni stresses that increase drag. This question is addressed for a three-dimensional laminar flow in a plane periodic channel with SHSs along both walls, in the presence of soluble surfactant. We consider the regime in which bulk diffusion is sufficiently strong for concentration gradients normal to the SHSs to be small. Seeking solutions that are periodic in the streamwise and spanwise directions, and exploiting a long-wave theory that accounts for rapid transverse Marangoni-driven flow and shear-dispersion effects, we thereby reduce this high-dimensional problem to a one-dimensional model for the surfactant distribution. The system exhibits multiple regimes where asymptotic solutions can be constructed, which compare favourably with numerics. Some are characterised by advection and diffusion-dominated processes, where the liquid-gas interface exhibits shear-free behaviour and the DR is at its maximum. In contrast, others are dominated by Marangoni effects, where the liquid-gas interface exhibits no-slip behaviour and the DR vanishes. This analysis provides a guide for designing surfactant-contaminated SHSs to maximise the DR for applications.